All Solutions - What's the best angle? - April 29 - May 3, 1996
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Correct solutions were submitted by:
Cygnet, Tasmania, Australia Amy Forster, age 11, Home schooled Fairfield High School, Fairfield, Connecticut Bilal Seyal, Grade 9 Franklin County High School, Rocky Mount, Virginia Rita Beckner, Grade 9 Garfield High School, Seattle, Washington Ben Warfield, Grade 12 Granada High School, Livermore, California Ethan Castor, Mike Sue, Grade 10 Hanover High School, Hanover, New Hampshire Carson Henry, Grade 9 Lakeside School, Seattle, Washington Joel Stonington, Sydney, Mahri, and Blythe, Carley, Katy, Vanessa, and Peter, Grade 9 Georgetown Day School, Washington, DC Ale Borensztein, David Ain, George Crowley, Josh Hersh, Leah Rinaldi, Nishant Kumar, Phoebe Stone, Jon Simon, Grade 8 Right but not general: Cheshire High School, Cheshire, Connecticut Rosina Pannone, Unknown, Caryl Anquillare, Grade 10 Georgetown Day School, Washington, DC Daniel Myers, Doug Rosenthal, Grade 8 J.P. Taravella High School, Florida Brent Tworetzky, Grade 9
From: Carson Henry Grade: 9 School: Hanover High School Answer: The person who said that there are more seats with equally good lines of sight is right because if you draw a circle with a circle around it with home plate, first base, and the seat as the vertices and make it so the triangle has a 25 degree angle where the seat is, you can then move the third vertex, which is the seat, around the circle and all the seats will be equally good because your line of sight will always include home plate and first base. ********************************************* The second person, who said you could sit in more than one seat, is correct. Solution: I drew two points, Home plate and first base,on a piece of paper. I cut a piece of paper so that it was at a 25 degree angle. I lined the 25 degree angled piece of paper with the home plate and first base, and put points whereever it was 25 degrees.The points I drew formed two circles that intersected at home plate and first base, so you got a sort of figure of 8 of possible places to sit with a 25 degree angle between home plate and first base. Since spectators can't sit in the field the possible seats that give a 25 degree view are two arcs on opposite sides of the players within the sitting area (assuming spectators can sit 360 degrees round the players; otherwise there is only one arc of possible seating. From doing this problem I discovered there must be a relationship for circles where lines from each end of a chord to a point on the circumference of the circle make the same angle for any point on the circumference. Amy Forster,age 11, grade 7, Home schooling, Cygnet, Tasmania, Australia. Wilkins/Forster family Crooked Tree Point. Cygnet. Tasmania. Australia ********************************************* Ethan Castor Granada High School Livermore, CA Grade 10The Second Guy was right. There is an infinite number. ********************************************* Bilal Seyal, grade 9, Fairfield HS The second person is right because there is more than one place where the angle forms 25 degrees. Put the two points, home plate and first base, on a circle so that the arc formed by these two points is 50 degrees. An inscribed angle connecting the two points would be 25 degrees. An infinite number of inscribed angles could be formed that are 25 degrees and connect the two points, but since there aren't an infinite number of seats, it's safe to say that there is more than one. ********************************************* I think every point on this circle except the two lined up with first and home plate are 'best seats'. Also you can use the 100, 200 and 300 levels, to make literally hundreds of 'best seats'.
After further contemplation I think the circle should go through all three points, first, home and the best seat. -Joel ********************************************* Ale Borensztein Grade 8, Georgetown Day School
These are a couple that would work, but it will work for any inscribed angle of arc GKH. ********************************************* David Ain Grade 8, Georgetown Day School
Every seat in the row where one seat has a line of sight of 25 degrees has a line of sight of 25 degrees. ********************************************* George David Crowley Grade 8, Georgetown Day School
The reason is simple. Side Angle is not a proof for congruency. ********************************************* Josh Hersh Grade 8, Georgetown Day School
All inscribed angles are half of their intercepted arcs. ********************************************* Leah Rinaldi Grade 8, Georgetown Day School
The second friend was right. There will be more than one seat where the angle of the line of sight is 25 degrees. There will be an infinite number of seats. ********************************************* Nishant Kumar Grade 8, Georgetown Day School
Point A and B are home and first. Since all the inscribed angles intercept the same arc they are the same, all 25 degrees. ********************************************* Phoebe Stone Grade 8, Georgetown Day School
When you make the base-line between home plate and first base a chord of a circle which intercepts a 50 degree arc, there are numerous inscribed angles that can be drawn to the end-points of the chord. The measure of an inscribed angle is half of its intercepted arc, thereby making each of the inscribed angles 25 degrees. If each of the seats is a point to which an inscribed angle is drawn, there are multiple 25 degree angles that can be formed. ********************************************* Jon Simon Grade 8, Georgetown Day School
There is an infinite number of seats which are best. ********************************************* Rita Beckner grade nine Franklin County High School Rocky Mount, Virginia
There are many places to sit with 25-degree angles of sight to first base and home plate. ********************************************* Sydney / Mahri / Blythe
The second person was right in stating that there are many places you can sit in the stadium and still have an ideal seat. These seats can be any place on the circle (or upper levels from the circle) and they will have a 25 degree viewing angle. ********************************************* Carley, Katy, Vanessa, Peter
From this construction we have concluded that the ideal seat does not have one set point but is at any point on the circle, and at any level. ********************************************* I hope you don't tell the deal about the good seats to the Mariners management: they're always looking for ways to make the extra buck (or 8 million), and since there are a very large number of such seats (defined by the set of vertices of a triangle with a fixed opposite side of 90 feet and a vertex angle of 25 degrees), it might get a little rough on those of us who actually buy tickets there regularly if they decided to put a premium on them. I can also say that your basic assumption, at least for the Kingdome, is a crock: you are much better off with outfield seats out in the open than with perfectly angled seats under the roof on the 200 level. If you're planning on seeing one of the Cleveland games, you'd better call ahead for tickets, they're selling out. [Ben sent the following after I asked for an explanation of his answer.] Boston plays at Seattle on the 30 and 31st of May and the 1st and 2nd of June. I'll be in section 116, myself. The seats described are on the circumference of any circle constructed such that the circumference passes through home plate and first base and the central angle to the bases is 50 degrees. There are probably three sets of relevant circles, each consisting of a large number of cirles very close to each other (to adjust for the pitch of the bleachers), one for each deck, each circle having a radius of just over 106 feet (I'm not sure how far up the 300 level is). Having thought about it some, I am willing to concede that that is a pretty good measure of a seat, but I maintain that the circles that extend to the 200 level, while the angles to the bases may be great, do not provide a worthwhile baseball experience. I hope you're not a Twins fan: Randy Johnson should be pitching. Regards, Ben Warfield Garfield High Home of the 1995 AL West Champion Seattle Mariners ********************************************* From: Mike Sue Grade: 10 School: Granada The second guy is right. There are many places where you get this 25 degrees and they are all on 2 circles with the radius of approximately 106.5 ft. The circles are centered on the perpendicular bisector of the line segment between home and first approx. 96.5 ft. toward the pitcher mound, and the other 96.5 feet away from the pitcher mound. [Mike added the following explanation] Let Home to First be a line segment from H to F. Select a point V such that angle HVF = 25 degrees. Then construct a circumscribed circle which can be called O about triangle HVF. the measure of arc HF = 2 times the measure of angle HVF = 50. Let X be any point on arc HVF, such that X can't be H or F. The measurement of angle HXF = one half of the measurement of arc HF = 25. Thus the locus of vertices X forms the measurement of angle HXF = 25 = the arcHXF where H and F aren't included. To find the center of circle O: Assume the baseball diamond is a 90 ft square. Therefore the line segment HF = 90 ft. Construct a perpendicular bisector to HF intersecting H at M. This line passes through the center of circle o. The measurement of angle HOF = measurement of arc HF = 50. The measurement of angle HOM = one half the angle HOF = 25. HM = one half of HF = 45. R = 45/(sin 45) = 106.5 ft. OM = 45/(tan 45) = 96.5 ft. Arc HXF can be on either side of segment HF. Well there you go. I hope it's sufficient. ********************************************* First of all, J.P. Taravella is a school in Florida (it came third at the Mu Alpha Theta state convention). Secondly, the question asks whether or not a 25-degree angle between two points can be found in more than one place. # # # # FIRST BASE # * # HOME PLATE * The pathetic diagram I tried to use to explain is simply showing how a 25-degree angle can be achieved between two places at infinitely many points (an ellipse is formed, is it not?). In this particular question, seeing as seats at a stadium are in several places surrounding the field, a different 25-degree angle can be found at different seats, not just at one seat. Indeed, the smaller the seats are, the greater amount of 25-degree angle seats are available. Brent Tworetzky ********************************************* Rosina Pannone Grade 10 Cheshire High School Cheshire, CT The man who said that there is more than one seat where the lines of sight between home plate and first base meet at a 25 degree angle, is correct. In order to figure this out, I drew a diagram of a baseball diamond. I used the line between home plate and first base as the base of a triangle. Then, since the third angle had to equal 25 degrees, that left 155 degrees to split up between the two base angles. There are many different ways to split up 155 degrees between two angles, and each produce different vertices where a good seat is. I did five triangles, and arrived at five different seats, and there are even more. Therefore, there is more than one best seat. ********************************************* Unknown In doing this Problem of the Week, I discovered that there is more than one seat that you can sit in to be at a 25 degree angle between first base and home plate and still have the best seat. There are actually many places to sit where the angle would be 25 degrees. This would depend on how close to the field you were and how far to each side you were. For instance you could be sitting so that the triangle between you, home and first was 25, 80, and 75 degrees or 25, 65 and 100 degrees. The angle where you were sitting would still be 25 degrees. ********************************************* Caryl Anquillare Grade 10 Cheshire High School Cheshire, CT The person who said that there are more than one seats in the Stadium were you would be at a 25 degree angle between first and home plate was correct.This is because when you have a triangle with one set side length, there are many different ways you can make a triangle with its opposite angle equalling 25 degrees. ********************************************* Daniel Myers Georgetown Day School Washington, DC Martha's Class, Grade 8
The second person is right because you can make 25 degree angles out of the two lines that you draw from home and first in more than one place. The diagram shows this. With all of the seats that form 25-degree angles you can form an arc. ********************************************* Doug Rosenthal
All points on the arc are 25.
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